Fans



G; c. GROFF Jan. 16, 1968 FANS 5 Sheets-Sheet l File d March 2, 1967 INVENTOR. GERALD C. GROFF.

ATTORNEY.

G. c. GROFF 3,363,832.

FANS

5 Sheets-Sheet 2 Jan. 16, 1968 Filed March 2, 1967 R. F. mm mm. v 2. k N, 9 1 c M m 3 s fl Y B N wwm J gm $.34". w 2 i 223mm t n v v V m V m N w WW w x m K m N. I h .0? 0 m m m 9 N ATTORNEY.

Jan. 16, 1968 NTOR. GERALD C. GROFF.

A T T o R N E Y a United States Patent Ofl ice 6 Claims. (Cl. 230-134) ABSTRACT OF THE DISCLOSURE A centrifugal fan in which the inlet sections of the fan blades are shaped so that the means pressure gradient of the air passing through the fan is substantially normal thereto and the tip portions of the blades are skewed and overlap one another, the intervening blade sections having a gradual twist.

This application is a continuation-in-part of my copending application Ser. No. 481,585, filed Aug. 23, 1965.

This invention relates to a fan and, more particularly, to a centrifugal fan which provides improved aerodynamic and acoustic performance in ventilating and air conditioning use.

A centrifugal fan is usually considered as having an axial inlet and a radial outlet. Such fans are generally classified according to the inclination or curvature of their blades at the outlet as, radial, or backward or forward curved or inclined types. A backwardly curved bladed fan generally provides better aerodynamic efiiciency or performance but is noiser than a forwardly curved bladed fan because of blade passage frequency sound generated by the fan blades and intensified by the scroll cut-off. The radial bladed fan has found little general acceptance in the field of ventilation and air conditioning.

Generation of sound at the blade passage frequency is discussed in a paper entitled Experimental Study of Noise Reduction in Centrifugal Blowers by T. F. W. Embleton, The Journal of the Acoustical Society of America, volume 35, No. 5, May 1963, pp. 700705, and in patents noted therein including, British Patent No. 1,051 of 1885, and United States Patents Nos. 353,- 994, 1,017,215, and 2,160,666. Reference may be had to this material if a more complete understanding of the blade passage frequency sound phenomenon is desired. In brief, by properly skewing (i.e. tilting) the blade tips blade passage frequency sound can be effectively suppressed or eliminated. However, it has not been practical to skew the blade tips sufliciently to completely eliminate blade passage frequency sound because of a resultant adverse effect on aerodynamic performance.

Most fans are designed by following various rules of thumb or empirical experience to produce a fan suitable for the desired application and competitive in the field with other fans which are similarly designed. Centrifugal compressor design has generally received a more enlightened approach. An example of an analytical approach useful in such designs is found in an article, Crossover Systems Between the Stages of Centrifugal Compressors, by Gaylord 0. Ellis in Transactions of the ASME, Journal of Basic Engineering, vol. 82, 1960, pp. 155-167, in which a method is discussed for providing centrifugal compressor crossover vanes which produce a mean pressure gradient of the compressed fluid substantially perpendicular to the surface of the vanes throughout the passage of the fluid across the vanes. This compressor design expedient has been found to improve the aerodynamic performance of a fan impeller, but the considerations are different because, by definition, the fluid passed Patented Jan. 16, 1968 by a centrifugal compressor is compressed, but the fluid passed by a fan is effectively not compressed and fan impeller design includes an additional consideration of impeller rotation. The result is that fluid passing through a fan undergoes a severe diffusion process in which the flow is much more likely to separate from the surface of a blade than in a practical compressor design. Separation is a phenomenon in which the fluid passing through a fan ceases to follow the surfaces of the blades and eddies and other factors combine to reduce aerodynamic performance. This is a primary reason why arbitrary adjustment such as the previously mentioned skewing of the blade tips to suppress blade passage frequency sound has been detrimental to performance when applied to otherwise aerodynamically satisfactory fans.

It is a primary object of this invention to provide a new and improved fan and, more particularly, a centrifugal fan of improved aerodynamic and acoustic performance.

Another object is provision of a new and improved centrifugal fan having an impeller of such configuration at the fan inlet that the fluid passing through the fan is provided with a mean pressure gradient generally normal to the surfaces of the blades, with tips of the blades skewed or tilted for effectively eliminating blade passage frequency sound, the inlet portion and tip of each blade being interconnected by an intermediate portion which twists gradually while imparting adequate energy transfer to the fluid passing through the fan, thus providing improved aerodynamic and acoustic performance. A related object is sizing of the inlet to minimize the relative flow velocity for a given flow rate, thus minimizing velocity dependent noise.

These and other objects of the invention will be apparent from the following description and the accompanying drawings in which:

FIGURE 1 is a perspective view of a preferred embodiment of a centrifugal fan incorporating features of the invention;

FIGURE 2 is a perspective view of an impeller of the fan shown in FIGURE 1, with imaginary lines, planes and surfaces added and parts broken away and removed for more comprehensive and clearer illustration;

FIGURE 3 is a schematic illustration of the hub, shroud, inlet ring and a rotated projection on an axial plane of the profile of any one of the blades shown in FIGURE 2;

FIGURE 4 is a sectional view taken on the line IV-IV in FIGURE 3;

FIGURE 5 is a graph of blade tilt angle T against distance S/S along the mean flow surface of the blade, as projected on an axial plane;

FIGURE 6 is a log-log graph of pressure rise through the fan divided by tip diameter squared. P /D against the flow rate through the fan divided by tip diameter cubed Q/D and illustrates the applicable design range for the peak performance operating point of the fan; and

FIGURE 7 is a diagrammatic perspective view illustrating a feature of the impeller design.

Referring to the drawings, FIGURE 1 illustrates a fan having a housing 11 including an inlet ring 12, on a scroll 13 encasing an impeller 14 (FIGURE 2) mounted for normal clockwise rotation about an axis 15 (FIGURES 2, .3 and 7). The fan has an axial inlet 16 to the impeller 14 and a radial outlet 17 (FIGURES 2 and 3) from the impeller opening into the scroll 13 for passage of air past a scroll cut-off 18 and then into a scroll outlet 19. The impeller 14 may best be seen in FIGURE 2 and includes a hub 20 having a spinner 21 at its inlet 22, and a plurality of blades 23, as illustrated eleven blades, each secured to the hub 20 and to a shroud 24 concentric with the 3 hub at the inlet end 22 of the impeller. The hub 20 and shroud 24 may have any aerodynamically suitable configuration (FIGURE 3) as is understood in the art and within the limitations of the subsequent description.

The hub and shroud design must be of suitable aerodynamic configuration to provide an efiicient turn of the air flow from the axial to the radial direction and to provide a gradual deceleration of the relative flow velocity from the inlet 16 and 22 to the impeller outlet 17. The axial length of the shroud 24 will be in the range of 0.2 to 0.3 times the impeller diameter at the blade tips 25, with the actual shroud length usually dictated by design requirements.

In order to eliminate adverse aerodynamic effects of leakage inflow at the clearance between the impeller shroud 24 and the inlet ring 12 of the scroll collector 13, it has been found desirable to provide an extension 26 to the leading end of the shroud, as shown in FIGURES 2 and 3, so that the inlet ring 12 is slightly overlapped -by the shoud extension 26. The shroud extension is a minimum of .04 time the impeller tip diameter to be effective.

To provide improved acoustical performance and a minimum aerodynamic deceleration rate for the flow from the impeller inlet 22 to the outlet 17, it has been found desirable to provide an inlet flow area for the impeller such that the relative flow velocity is a minimum for a given design flow rate. From the following equation for inlet flow area, the inlet diameters (FIGURE 3) of the hub 27 (D (which will generally be specified by design considerations) and the shroud 28 (D in inches, as measured in an inlet plane (normal to the axis at the forwardmost part of the blades) may be computed:

'where:

Q is the design flow rate through the fan in c.f.m.;

N is the design rotational speed of the impeller 14, in

r.p.m.; and

F is a blade blockage factor equal to the ratio of the free area for the how of air at the impeller inlet 22 to the total annulus area at the inlet. This factor is preferably approximately 0.9.

The hub and shroud 24 together defining an annular flow passage from the inlet 22 to the outlet 17, and the blades 23 divide the annular flow passage into the plurality of side-by-side or blade-to-blade flow passages 29 each extending from the inlet to the outlet. The blades 23 are substantially equally spaced about the hub 20 and their tip edges are substantially parallel to each other and are skewed or tilted with respect to an axial plane 30 from the shroud 24 to the hub 20 in the direction of rotation of the impeller (herein clockwise as indicated by the arrow in FIGURE 2) so that the juncture 31 of the shroud and the tip of a blade falls substantially in the axial plane 30 (any plane in which the axis of rotation lies) which includes the juncture 32 of an adjacent tip with the hub, as may best be seen in FIGURE 2. During rotation of the illustrated embodiment, each tip 25 describes a surface of revolution generally parallel to the axis of rotation, herein a cylinder. The inlet portion 33 (FIGURE 3) of each blade, preferably between 25% and 40% along a mean flow surface, is so shaped that during normal operation, its surface is substantially perpendicular to the mean pressure gradient of the air passing where: L is in the range of 0.1 to 0.45; and

The interdependence of various factors used in the calculations is indicated in the following table which shows typical values of constants which may be used for corre sponding values of the factor H.

H Cu O; Y

The value of the factor M represents the fractional length along the mean flow surface 35 to the point of minimum tilt and is preferably selected to provide a gradual rate of change of tilt in the intermediate portion 34 of the blade and is near 0.40 if the tilt angle T at the impeller tip 25 (S/S =1.0) is less negative than at the inlet, or near 0.25 if the tilt angle at the tip is equal to or more negative than at the inlet. The tilt angle at the inlet is equal to G and the tilt angle at the tip is are tangent irD /Bb]. The number of blades used in an impeller will normally be between seven and nineteen, usually an odd number to avoid combination of blade interference noise generation with inlet strut wakes, and so forth, and is selected to provide a balance between good flow guidance in the intermediate portion of the flow passage and excessive blockage in the inlet portion of the flow passage. Splitters or partial blades (not shown) may be provided, if desired, according to the principles outlined herein.

The configuration of the blades may be considered on three dimensional polar coordinates R, 6, and Z. Theta for each point on the blade is measured from an axial plane through the mean flow surface 35 at the inlet or the leading edge 50 of the blade and is measured positive in the direction of rotation. Z is the length along the axis and R is the radius from the axis.

The coordinate theta (6) for any station (S/S along the blade intersection with the mean flow surface is obtained from a mean How angle beta (,8) (FIGURE 7), measured on the mean flow surface 35 between the mean flow direction (arrow 51) at the desired station (S/S 52 and the rotational projection (arrow 53) of this direction on an axial plane containing the station. The flow angle, 6, may be prescribed for the blade intersection with the mean flow surface or calculated from prescribed relative flow velocities and whirl changes as disclosed in the previously noted publication by Ellis, or by other procedures known in the art. The mean flow angle may be modified, usually in the inlet and outlet regions of the impeller, to allow for incidence and flow deviation, or slip. The theta coordinates for the blade 23 and mean how surface 35 intersection may be calculated for each station, as 52, along the axial plane projection from the formula:

The theta coordinates for the intersection of the blade with the hub and shroud are determined by the blade tilt angle T, calculated as described above. For each station (S/S along the mean flow surface, the difference between theta for the mean flow surface and blade intersection, and the hub and blade intersection, or shroud and blade intersection is obtained graphically (or by calculation) on a transverse plane (FIGURE 2) through that station. The line of blade intersection with the transverse plane is extended in the transverse plane from the mean flow surface to hub and shroud, at the angle T, for that station, although some adjustment from a straight line may be made to achieve practical junctures with the hub and shroud, as previously noted.

The selection of a particular value of H is a design variable depending on the particular or limiting speed of rotation for the desired pressure rise through the fan, and is the product of the static efficiency and tangential component of the absolute air velocity at the impeller tip 25, divided by the impeller mean tip speed in feet per minute.

G 1065- are tangent where:

X must be in the range of 0.90 to 1.10;

C must be in the range of 0.3 to 0.45, higher values corresponding to higher values of H; 1

R is the radius from the axis to the mean flow plane at in inches; and R is the radius of curvature of the mean flow surface at in inches;

where:

M is in the range of 0.25 to 0.40;

B is the number of blades;

b is the distance in an axial plane and parallel to the axis of rotation 15, between the hub 20 and the shroud 24 at the tip 25, in inches, and may be calculated from where with reference to FIGURE 3):

2/3Nl/3 D920 2 G are tangent where R is the radius from the axis 15 to the mean flow surface 35 at the inlet plane, in inches; and

R is the radius of curvature of the mean fiow surface 35 at the inlet plane, in inches.

Note that throughout this calculation the values for the various radii and the Ss may be determined graphically or calculated from an analytical expression of the impeller geometry.

F is the blockage factor, usually about 0.9;

Q is the design flow rate through the fan in c.f.m.;

D is the tip diameter taken at the mean flow surface, in

inches;

N is the design rotational speed of the impeller in r.p.m.

and for standard air equals:

where P is the design static pressure rise for the fan in inches of water, gage;

H must be in the range of 0.35 to 0.60; and

C must be in the range of 0.2 to 0.3, higher values corresponding to higher values of H;

through this portion of the fan. As shown in FIGURE 5, an intermediate portion 34 (FIGURE 3) of each blade between the inlet portion 33 and the tip 25 is so shaped as to join the inlet portion and the tip with a gradual change in tilt throughout the intermediate portion.

The annular flow passage may be divided into imaginary equal volume portions by an imaginary mean flow surface of revolution 35 (FIGURES 2, 3 and 7) extending from the inlet to the tip so that the area of the annular flow path between the blades 23 is the same above and below the mean flow surface 35 when measured in a transverse plane, as 36 (FIGURE 3), normal to the mean flow surface 35.

For a clearer understanding in describing blade tilt (which is the same as skew at the tip), the letters A and B will be added to reference numerals. The letter A refers to tilt at an intermediate portion of a blade, and the letter B refers to tilt (or skew) at the tip 25. The tilt angle T of any point 38A, 38B along the intersection of a blade 23A, 23B and the mean flow surface 35, by definition, lies in an imaginary transverse plane 39A, 39B normal at the point 38A, 38B to the mean flow surface 35. The mean flow surface 35 intersects the blade 23A, 23B on a line 40A, 40B, and the transverse plane on a line 41A and not shown for blade 23B. The tilt angle T (in the transverse plane 39A, 39B) is the acute angle between the tangent line 42A, 42B to the blade intersection 43A, 43B with the transverse plane 39A, 3913; an an axial plane 44A, 448 through the point 38A, 38B. The tilt of the blade for all points in any particular transverse plane, as 39A, 39B, is substantially the same but with suit-able deviation 45 (FIGURE 4) at the hub 20 and 46 at the shroud 24 to provide practical junctures with the hub and shroud.

Within design limitations, as illustrated in FIGURE 6 between the parallel inclined lines 47 and 48, the angles T for any portion of a blade 23 may be deter-mined according to the following equation:

The following table shows typical values for the blades illustrated in FIGURE 2.

8 viewed from the inlet on any said transverse plane between the tangent to a blade intersection with the last said Z 6 R R R Distance 9 Mean 9 Hub Mean Shroud in Inches Hub Flow Shroud Tip in Inches Flow in Inches Surface Surface The surfaces of the blades 23 are smoothly contoured in the vicinity of their inlets 22 and tips 25 to match the calculated relative flow direction and allow for flow deviation from the blades, as is understood in the art, the calculated relative flow direction being based on the design flow rate and pressure rise for the impeller.

While a preferred embodiment of the invention has been described and illustrated, it should be understood that the invention is not limited thereto but may be otherwise embodied within the scope of the following claims.

I claim:

1. In a centrifugal fan, an impeller mounted for rotation about an axis, said impeller having a substantially axial inlet and a substantially radial outlet and including a hub and a shroud concentrically spaced about the hub and extending from the inlet to the outlet, and said impeller including means comprising a plurality of blades for propelling air through the fan with the mean pressure gradient of the air substantially normal to the surfaces of inlet portions of the blades extending from said inlet to intermediate portions of the blades at least one quarter the axial length of the blade surfaces from said inlet, said blades each extending between the hub and shroud and from said inlet to a tip edge at said outlet, the tip edges extending from the shroud to the hub and the juncture of the shroud and the tip of a blade falling substantially in an axial plane including the juncture of the hub with the tip of the adjacent blade in the direction of rotation of the impeller, and said inter-mediate portions extending from said inlet portions to said tip edges and twisting gradually therebetween.

2. The fan of claim 1 wherein said blades are substantially equally spaced about said axis and said inlet portions extend from said inlet up to 40% of said length.

3. A centrifugal fan comprising an impeller having a substantially axial inlet and a substantially radial outlet, an axis of rotation generally coincident with said axial inlet, means defining an annular flow passage from the inlet to the outlet and including a plurality of blades substantially equally spaced about the axis and each extending from a leading edge transverse to said axis at the inlet to a tip edge at the outlet, the tips being substantially parallel to each other and skewed in the normal direction of rotation with an end of the tip of a blade nearest said inlet falling substantially in an axial plane including an end of an adjacent tip farthest from said inlet, an imaginary mean flow surface of revolution dividing said annular flow passage into two portions of equal area measured on any imaginary transverse plane normal to said mean flow surface, and surfaces of said blades being tilted with respect to said mean flow surface at acute angles T measured positively in said direction of rotation transverse plane and an axial plane through said intersection substantially in keeping with the following equa- S is the length from the inlet to any said transverse plane taken along the mean fiow surface as projected on an axial plane through the intersection of the leading edge and the mean flow surface, in inches,

S is S from the inlet to the tip, in inches; and

the GS are constants determined as follows:

G -arc tangent (11.8X 10 flyl where:

P is the design static pressure rise for the fan in inches of water, gage; and

H must be in the range of 0.35 to 0.60;

C must be in the range of 0.2 to 0.3, higher values corresponding to higher values of H; and

F is a blockage factor, usually about 0.9;

G 10Zv 10 are tangent Y must be in the range of 0.04 to 0.065, higher values corresponding to higher values of H;

E is an angle in an axial plane between the axis of rotation and the tangent to the mean flow surface in the last said axial plane at X must be in the range of 0.90 to 1.10;

C must be in the range of 0.3 to 0.45, higher values corresponding to higher values of H;

R is the radius from the axis to the mean flow surface and R is the radius of curvature of the mean flow surface at 1 71'Dg l:3M ale tangent where:

M is in the range of 0.25 to 0.40; B is the number of blades; b is the distance in an axial plane and parallel to the axis of rotation, between opposite ends of the tips, in inches, and may be calculated from L7r D N where: L is in the range of 0.1 to 0.45; and

4. The fan of claim 3 and a shroud extending from the inlet to the outlet, a hub generally concentric within said shroud at said inlet and concentric with said axis, said hub being mounted for rotation about said axis and with said shroud defining said annular flow passage from the inlet to the outlet, said blades being mounted on said hub and each extending from the hub toward the shroud and therebetween from said leading edge to said tip edge, and said hub and shroud together with adjacent blades dividing said annular flow passage into a plurality of sideby-side flow passages each extending from said inlet to said outlet.

5. The fan of claim 4 wherein the characteristics of said inlet are defined by the following equation:

DSLDFZE where:

D is the impeller inlet diameter at the shroud, in inches;

and

D is the impeller inlet diameter at the hub, in inches.

6. A centrifugal fan impeller comprising an axial inlet including a hub and a shroud and therebetween blades, said inlet being defined by the following equation:

4, 2 1/3 D, -D =288 where:

D is the impeller inlet diameter at the shroud, in

inches; D is the impeller inlet diameter at the hub, in inches; Q is the design flow rate through the fan in -c.f.m.; N is the design rotational speed of the impeller, in

r.p.m.; and F is a blade blockage factor equal to the ratio of the free area for the flow of air at the inlet to the total annulus area at the inlet. This factor is preferably approximately 0.9.

References Cited UNITED STATES PATENTS 1,161,926 11/1915 Criqui 230134.45 1,341,882 6/1920 Criqui 230-134.45 1,781,165 ll/1930 Criqui 230-134.45 2,160,667 5/1939 McMahon 230-1341 2,441,411 5/1948 Hagen 230--134.45 2,484,554 10/1949 Concordia et al. 230-13445 2,548,465 4/1951 Burdett et a1. 230134.1 2,819,012 l/1958 Atkinson 230-134.45 3,028,140 4/1962 Lage 230-l34.45 HENRY F. RADUAZO, Primary Examiner.

UNITED STATES PATENT OFFICE CERTIFICATE OF CORRECTION Patent No. 3,363,832 January 16 1968 Gerald C. Groff It is hereby certified that error appears in the above numbered patent requiring correction and that the said Letters Patent should read as corrected below.

Column 3, line 20, for "time" read times line 67, after "passing" insert the following:

through this portion of the fan; As shown in FIGURE 5 an intermediate portion 34 (FIGURE 3) of each blade between the inlet portion 33 and the tip 25 is so shaped as to join the inlet ortion and the tip with a gradualchange in tilt throug out the intermediate-portion.

The annular flow passage may be divided into imaginary equal volume portions by an imaginary mean flow surface of revolution 35 (FIGURES 2, 3 and 7) extending from the inlet to the tip so that the area of the annular flow path between the blades 23 is the same above and below the mean flow surface 35 when measured in a transverse plane, as 36 (FIGURE 3] normal to the mean flow surface 35 For a clearer understanding in describing blade tilt (which is the same as skew at the tip) the letters "A" and "B" will be added to reference numerals. The letter "A" refers to tilt at an intermediate portion of a blade, and the letter "B" refer to tilt (or skew) at the tip 25; The tilt angle T of any point 38A, 388 along the intersection-of a blade 23A, 23B and the mean flow surface 35, by definition, lies in an imaginary transverse plane 39A, 39B" normal at the point 38A, 38B to the mean flow surface 35 The mean flow surface 35 intersects the blade 23A, 23B on a line 40A", 40B, and the transverse plane on a line 41A and not shown for blade 233. The tilt angle T (in the transverse plane 39A, 39B) is the acute angle between the tangent line 42A, 42B to the blade intersection 43A, 43B with the transverse plane 39A, 39B; and an axial plane 44A, 448 through the point 38A, 383. "The tilt of the blade for all junctures with the hub and shroud.

Within design limitations, as illustrated in FIGURE 6 betweez the parallel inclined lines 47 and 48, the angles T for any where (with reference to FIGURE 3) S is the length from the inlet plane to any transverse plane, as 36, taken along the mean flow'surface 35, as projected on an axial plane 49 through theintersection of the inlet edge 50 of the blade 23 and the mean flow surface, in inches;

8 is S from the inlet edge 56' fhe tip 25;

the G's are constants determined as follows:

G =arc tangent l (EN 3 1 (D2) 2 where:

R is the radius from the axis I5 to the mean flow surface 35 at the inlet plane, in. inches; and R is the radius of curvatureelitheneanilnmsuriace 35 at the inlet plane in inchesf Note that throughout this calculation the alues for the various radii and the S's may be determined graphically or calculated'from an analytical expression of the impeller geometry.

F is the blockage" "factor, usually akout 0.9;

Q is the design flow rate through't e fan in c.f.m.

D is the tip "diameter" taten'at-theI'mean-'f'low' surface, in inches;

N is the design rotational speed of, the impeller in r.p.m. and for standard air equals:

n n where:

P is the design static pressure rise for the fan in inches of water, gage;

H must be in the range of 0.35 to 0.60; and

C must be in the range of 0.2 to 0.3, higher values corresponding to higher values f H,

The selection of a particular value of H is a design variable depending on the particular or limiting speed of rotation for the desired pressure rise through the fan, and is the product of the static efficiency and tangential component of the absolute air velocity at the impeller tip 25, divided by the impeller mean tip speed in feet per minute.

G1= -l0G -l0 arc tangent I n 2 2 R 2Q 2/3 4 z 3 2 2 cos n -x 2 1 [Q D2 x SZKZF z/s l/s C where:

Y must be in the range of 0.04 to 0.065, higher values corresponding to higher values of H;

E is an angle in an axial plane between the a is of rotation and the tangent to the mean flow surface in the last mentioned axial plane at X 1 nust be in the range of 0.90 to 1.10;

C 'must'be in the range of 0.3 to 0.45, higher values corresponding to higher values of H;

R is the radius from the axis to the mean flow plane at 2 in inches; and R is the radius of curvature of the mean flow surface at 2 in inches;

"D 2 1 EM arc tangent Z -3MG G1 [3M -l) SM-Z Rh 0 M where:

M is in the range of 0.25 to 0.40;

B is the number of blades;

b is the distance in an axial plane and parallel to the axis of rotation 15, between the hub 20 and the shroud 24 at' the tip 25 in inches, and may be calculated from 2 2 1m D N column 4, line 71, beginning with "The selection" strike out allto'and including "theimpeller" in line 75, same column 4; columns 5 and 6 should be deleted in their entireties; column 8, lines 67 to 71, for that portion of the formula reading 2 2 D D read 4 24 Signed and sealed this 17th day of June 1969.

(SEAL) Attest:

EDWARD M.FLETCHBR,JR. WILLIAM E. SCHUYLER, JR. Attesting Officer Commissioner of Patents 

